Unducted propulsion system

ABSTRACT

Apparatuses and systems are provided herein for unducted propulsion systems. The system includes an aft housing for low drag for high subsonic sustained flight. A plurality of blades are affixed to the aft housing, wherein the housing defines a flowpath curve extending from the axial extent of the aft blade root to the aft end of the aft housing. The flowpath curve is described by an axial direction parallel to an axis of rotation and a radius from the axis of rotation. The flowpath curve includes first point having a first radius where the radius reaches a maximum aft of the aft blade root and a second point forward of the first point having a second radius where the radius stops decreasing. The ratio of the first radius to the second radius is greater than or equal to 1.081.

TECHNICAL FIELD

The technology relates generally to an unducted propulsion system.

BACKGROUND

Generally, a fan of an aircraft propulsion system produces thrust byaccelerating air passing through the fan. Factors that are detrimentalto efficiency of thrust production include losses in energy in air as itenters and passes through the fan, velocity contributions that do notcontribute to thrust (such as swirl and vortices in the air leaving thefan), frictional drag forces on external surfaces of an aircraftpropulsion system, and shockwave-related drag forces (e.g., wave drag)on external surfaces of the aircraft propulsion system. Thus, for anaircraft propulsion system, the goal is to generate a given amount ofthrust without requiring excessive input power to the fan. As such, itis desirable to minimize inefficiency in the production of thrust.

BRIEF DESCRIPTION OF THE DRAWINGS

Disclosed herein are embodiments of systems and apparatuses pertainingto an unducted propulsion system. This description includes drawings,wherein:

FIG. 1 shows an elevational cross-sectional view of an exemplaryunducted propulsion system having an axis of rotation, forward and aftblade assemblies, forward and aft housings, engine inlet and engineexit, in accordance with some embodiments;

FIG. 2 is a schematic, perspective view of an exemplary gas turbineengine attached to a wing of an aircraft in accordance with someembodiments;

FIG. 3 is a cross-section of an exemplary unducted propulsion systemshowing curvature along a flowpath curve in accordance with someembodiments;

FIG. 4 illustrates a flow of air through an assembly of blades of anunducted propulsion system in accordance with some embodiments;

FIG. 5 illustrates an effect on air when air moves over a non-linearsolid surface;

FIG. 6 shows a schematic illustration of three surface locationsdefining an exemplary flowpath curve for an aft housing in accordancewith some embodiments;

FIG. 7 illustrates examples of flowpath curves for an aft housing inaccordance with some embodiments;

FIG. 8 shows an exemplary graph of the same three flowpath curves inFIG. 7 in terms of their first derivative with respect to axial distancein accordance with some embodiments;

FIG. 9 illustrates curvature by showing the second derivative withrespect to axial distance of the three curves in FIG. 7 in accordancewith some embodiments;

FIG. 10 shows the same elevational cross-sectional view of the unductedpropulsion system of FIG. 1 , but with element numbering referring inparticular to a forward housing or spinner portion, in accordance withsome embodiments;

FIG. 11 is a chart 200 depicting shapes for a forward housing of anunducted propulsion system, according to some embodiments;

FIG. 12 is a chart 1200 depicting bounds on the shape of the forwardhousing of an unducted propulsion system, according to some embodiments;and

FIG. 13 is a flow chart of a method of operating an unducted propulsionsystem, according to some embodiments.

Elements in the figures are illustrated for simplicity and clarity andhave not necessarily been drawn to scale. For example, the dimensionsand/or relative positioning of some of the elements in the figures maybe exaggerated relative to other elements to help to improveunderstanding of various embodiments of the present disclosure. Also,common but well-understood elements that are useful or necessary in acommercially feasible embodiment are often not depicted in order tofacilitate a less obstructed view of these various embodiments of thepresent disclosure. Elevational cross-sectional views of an unductedpropulsion system in the figures depict external flowpath curves formedby the intersection of the external surface of the housings with a planethat includes the axis of rotation. Such cross-sectional views alsoindicate structures such as blades that facilitate understanding of theembodiments of the present disclosure. Limiting the section view to asingle side of the axis of rotation does not imply that the system isaxisymmetric about the axis of rotation. The cross-sectional views areused to illustrate certain characteristics, for example the shape of ahousing associated with a blade assembly. Also, the drawings omitcertain details in the system not needed to fully appreciate certainaspects of the system. Certain actions and/or steps may be described ordepicted in a particular order of occurrence while those skilled in theart will understand that such specificity with respect to sequence isnot actually required. The terms and expressions used herein have theordinary technical meaning as is accorded to such terms and expressionsby persons skilled in the technical field as set forth above exceptwhere different specific meanings have otherwise been set forth herein.

DETAILED DESCRIPTION

Aspects and advantages of the present disclosure will be set forth inpart in the following description, or may be obvious from thedescription, or may be learned through practice of the presentdisclosure.

Reference will now be made in detail to present embodiments of theinvention, one or more examples of which are illustrated in theaccompanying drawings. The detailed description uses numerical andletter designations to refer to features in the drawings. Like orsimilar designations in the drawings and description have been used torefer to like or similar parts of the invention.

The word “exemplary” is used herein to mean “serving as an example,instance, or illustration.” Any implementation described herein as“exemplary” is not necessarily to be construed as preferred oradvantageous over other implementations.

As used herein, the terms “first”, “second”, and “third” may be usedinterchangeably to distinguish one component from another and are notintended to signify location or importance of the individual components.

The terms “forward” and “aft” refer to relative positions within a gasturbine engine or vehicle, and refer to the normal operational attitudeof the gas turbine engine or vehicle. For example, with regard to a gasturbine engine, forward refers to a position closer to an engine inletand aft refers to a position closer to an engine exit or exhaust.

The terms “upstream” and “downstream” refer to the relative directionwith respect to fluid flow in a fluid pathway. For example, “upstream”refers to the direction from which the fluid flows, and “downstream”refers to the direction to which the fluid flows.

The terms “coupled,” “fixed,” “attached to,” and the like refer to bothdirect coupling, fixing, or attaching, as well as indirect coupling,fixing, or attaching through one or more intermediate components orfeatures, unless otherwise specified herein.

The term “propulsive system” refers generally to a thrust-producingsystem, which thrust is produced by a propulsor, and the propulsorprovides said thrust using an electrically-powered motor(s), a heatengine such as a turbomachine, or a combination of electric motor(s) andturbomachine.

The term “housing” refers to a casing that encloses a propulsion systemand provides an aerodynamic exterior. A housing may be comprised of orinclude a hub, spinner and nacelle. In addition, a housing may be eitherrotating about the axis of rotation or stationary, or segmented axiallyso that a portion is rotating while another portion is stationary.

The singular forms “a”, “an”, and “the” include plural references unlessthe context clearly dictates otherwise.

For a flowpath curve corresponding to a housing external surface, axialdirection, “z”, is parallel to the axis of rotation and radius, “r”, isthe distance from the axis of rotation. And an z-r plane is located byan angular coordinate theta (i.e., cylindrical coordinate system isadopted, where the coordinate theta precisely locates the orientation ofan z-r plane in 3D space). Because the housing external surface may notbe axisymmetric about the axis of rotation, the shape of the flowpathcurve may depend on the z-r plane used to define it. In thespecification and claims, in addition to stipulating that it includesthe axis of rotation, a z-r plane used to define the flowpath curve alsoincludes a point on a blade root within the blade assembly nearest to orassociated with the housing described by the curve. Furthermore, for anyaxial location, z, along the curve for which a housing is rotating aboutthe axis of rotation, rather than referring to a radius on the flowpathcurve at a specified z-r position with respect to the axis of rotation,the radius, r, is an “effective” radius for a housing cross-sectionalarea perpendicular to the axis of rotation at that axial, z, location.Thus, for axial locations where the housing is rotating, radius, r, isthe radius of a circle having the same cross-sectional area of thehousing in a planar section perpendicular to the axis of rotation.

The term “bulge” refers to the location on the flowpath curve where,proceeding along the curve away from the nearest/associated bladeassembly (i.e., forward of the forward blade assembly for the forwardhousing and the aft of the aft blade assembly for the aft housing), theradius reaches a maximum.

The term “local minimum” refers to the first location on the segment ofthe flowpath curve, proceeding from the bulge toward and through theaxial extent of the associated blade root, where the radius stopsdecreasing. If the radius monotonically decreases from the bulge throughthe axial extent of the associated blade root, then the local minimum isat the location on the segment of the flowpath curve farthest from thebulge. Thus, the local minimum is the nearest minimum radius location tothe maximum radius location that is also within the axial extent of theblade root or between the blade root and the maximum radius location. Itis understood that any gaps or steps in the flowpath curve arising fromconnecting, mating, or relative motion between components of the housingare ignored when determining the local minimum.

Approximating language, as used herein throughout the specification andclaims, is applied to modify any quantitative representation that couldpermissibly vary without resulting in a change in the basic function towhich it is related. Accordingly, a value modified by a term or terms,such as “about”, “approximately”, and “substantially”, are not to belimited to the precise value specified. In at least some instances, theapproximating language may correspond to the precision of an instrumentfor measuring the value, or the precision of the methods or machines forconstructing or manufacturing the components and/or systems. Forexample, the approximating language may refer to being within a 1, 2, 4,10, 15, or 20 percent margin.

Here and throughout the specification and claims, range limitations arecombined and interchanged, such ranges are identified and include allthe sub-ranges contained therein unless context or language indicatesotherwise. For example, all ranges disclosed herein are inclusive of theendpoints, and the endpoints are independently combinable with eachother.

The technology described herein relates to an unducted propulsionsystem, particularly the shape an external surface of one or morehousings encasing a propulsion system, for which housings can becomprised of a spinner, hub and/or nacelle.

A turbofan engine operates on the principle that a central gas turbinecore drives a bypass fan, the fan being located at a radial locationbetween a fan duct and the engine core. An unducted propulsion systeminstead operates on the principle of having the bypass fan locatedoutside of the engine nacelle. This permits the use of larger fan bladesable to act upon a larger volume of air than for a turbofan engine, andthereby improves propulsive efficiency over conventional engine designs.

Unducted propulsion systems may take the form of a propeller system, asused on a wide range of aircraft, e.g., radio controlled modelairplanes, drones, piston engine propeller aircraft, turboprop regionalaircraft, and large turboprop military transports. Another type ofunducted propulsion system, sometimes referred to as “open rotor”,consists of two blade assemblies, one in a forward position and one inan aft position, in which at least one of them rotates about an axis todeliver power to the propulsive stream that generates thrust. Such twoblade assembly systems offer some advantages, but also some challengesand are far less common than single blade row systems. As used herein,the term “propeller” may refer to the single blade assembly of anunducted propulsion system or the forward blade assembly of an unductedpropulsion system comprised of two blade assemblies. The term “fan” mayrefer to the either a propeller or both blade assemblies of an unductedpropulsion system.

According to the disclosure, an unducted propulsion system can enablehigh subsonic cruise flight speed. Cruise is a phase of the flight thatoccurs when the aircraft levels to a set altitude after a climb andbefore it begins to descend. Thus, as used herein, cruise represents acontinuous, high speed, and stable condition of flight for which anaircraft is intended to operate. This description is to distinguishcruise from certain conditions that are abnormal or transient, such asdive, in which the aircraft can reach high flight speeds, but theaircraft is not intended to experience for a substantial portion of themission from takeoff to landing.

An unducted propulsion system that enables highest subsonic cruiseflight can have two blade assemblies positioned in aerodynamicrelationship to one another. As used herein “aerodynamic relationship”means they are positioned such that one is downstream of the other so atleast a portion of the air acted upon by the forward blade assembly issubsequently acted upon by the aft blade assembly. This allows thetangential velocity, also known as swirl, imparted to the air by theforward blade assembly to be counteracted, i.e., at least partiallycanceled, by the change in tangential velocity imparted by the aft bladeassembly. At least one of the blade assemblies is a rotating assemblycarrying an array of airfoil blades that rotate about an axis ofrotation and are located outside the engine nacelle. The other bladeassembly may be another rotating blade assembly (rotor) or it may be astationary blade assembly (stator). Without the aft blade assembly tocancel the swirl of the forward blade assembly, the high power per unitfrontal or annular fan area required for high speed flight would leaveexcessive swirl in the air that passes through the unducted propulsionsystem, resulting in poor efficiency in producing thrust. For thisreason, single propeller propulsion systems, such as propellers onturboprop engines, typically power aircraft that do not exceed a cruiseMach number of 0.72.

If the unducted propulsion system is comprised of two blade assembliesthat are both rotors, the blades of the forward and aft blade assembliesare arranged to rotate about a common axis in opposing directions andare axially spaced apart along that axis. For example, the respectiveblades of the forward rotor assembly and aft rotor assembly may beco-axially mounted and spaced apart, with the blades of the forwardrotor assembly configured to rotate clockwise about the axis and theblades of the aft rotor assembly configured to rotate counter-clockwiseabout the axis (or vice versa).

If one of the two blade assemblies is a stator, this blade assembly doesnot rotate about an axis and is placed either aerodynamically upstreamor downstream of the rotating blade assembly, being the forward or aftblade assembly, respectively. If placed aerodynamically upstream of therotating blade assembly, the stationary blade assembly impartstangential velocity to the air in the direction opposite to thedirection of rotor rotation, referred to as counter-swirl. Because ofthe direction of rotation, the aft rotating blade assembly imparts achange in tangential velocity to the air to reduce the magnitude of thetangential velocity of the air that passes through it. If positionedaerodynamically downstream of the rotating blade assembly, thestationary blade assembly imparts a change in tangential velocity thatis opposite to the direction of tangential velocity imparted by therotor, referred to as de-swirl. By de-swirling the air that it receivesfrom the rotating blade assembly, the aft blade assembly reduces themagnitude of the tangential velocity of the air that passes through it.The blades in a stator are often referred to as “vanes”. However, thegeneral term “blade” and “blade assembly” are used herein to be used ineither a rotating blade assembly or stationary blade assembly.

For a stationary blade assembly, the aircraft structure may beintermingled, integrated, or merged with the blade assembly. Forexample, the pylon used to mount an engine to an aircraft may occupysome of the same axial extent along the rotating blade assembly axis ofrotation as at least some of the blades in the stationary bladeassembly. Also, portions of the aircraft structure may be designed toserve the purpose of counter-swirl for a forward blade assembly orde-swirl for an aft blade assembly. Thus, aircraft structures mayaugment or even replace some blades in a stationary blade assembly.

As used herein, the locations or coordinates indicated by distanceparallel to the axis of rotation and perpendicular to the axis ofrotation define the external flowpath surfaces of the indicatedstructure. The external flowpath surfaces work with the blade assembliesto affect the flow of the working fluid, typically air, through the fan.The external flowpath surfaces formed by one or more housings separatethe air stream accelerated by the fan from internal mechanisms,machinery, or equipment associated with the propulsion system. As theflight Mach number and acceleration of air through the fan increase, theshapes of these external flowpath surfaces become increasingly importantto avoid high pressure loss or drag. In addition, these externalflowpath surfaces may bulge, i.e., increase in size, axially away fromthe vicinity of the blade assembly to accommodate above-mentionedinternal items.

For an unducted propulsion system, high speed flight requires evenhigher velocity through the fan and over the flowpath surfaces formed bythe one or more housings. As used herein, “fan stream” is the fluidstream accelerated by the fan to produce thrust. Such velocities mayreach or exceed the speed of sound, or Mach 1. Under certain conditions,high Mach flows generate dramatic increases in pressure loss and drag,penalizing the thrust producing performance, or efficiency, of thesystem. This may lead to poor fuel efficiency. Also, it may be desirableto limit the diameter of the fan to avoid penalties associated withweight, drag, and installation on an aircraft. However, a compact fanresults in high thrust per unit frontal, or annular, area of the fanand, thus, higher acceleration than if the fan diameter were not soconstrained. Furthermore, axial length of the system, and thus, thelength of the flowpath surfaces that bound the fan stream, contributesto drag and weight. At the same time, reducing the axial length can alsopenalize performance of the unducted propulsion system by causing highmagnitudes of flowpath surface curvatures, resulting in regions of highMach number. Accordingly, it is desirable to provide an unductedpropulsion system with an external flowpath shape of a housing upstreamof and within the axial extent of a forward blade assembly that enablesthe aircraft to fly at high subsonic speeds with good efficiency andwith transonic flow within the fan. It is also desired to provide anunducted propulsion system with an external flowpath shape of a housingdownstream of and within the axial extent of an aft blade assembly thatenables the aircraft to fly at high subsonic speeds with low loss anddrag.

According to the disclosure, an unducted propulsion system for asubsonic aircraft having a cruise Mach number, M₀, 0.74 or greater, forexample, 0.74<M₀<0.86 or between cruise Mach number 0.78 and 0.84, hasof an axis of rotation, a forward blade assembly, an aft blade assembly,a forward housing, and an aft housing. The forward and aft bladeassemblies each include a plurality of blades, each blade having a rootproximal to the axis of rotation and a tip distal from the axis ofrotation. A flowpath curve corresponds to the intersection of the afthousing external surface with a plane containing the axis of rotationand the aft-most point of an aft blade root. For the flowpath curve,axial direction, z, is parallel to the axis of rotation, increasing inthe aft or downstream direction. For the flowpath curve, radialcoordinate, r, is distance from the axis of rotation.

The flowpath curve has a bulge and local minimum. The bulge locationwith radius r_(b) is found by proceeding aft from the aft-most point ofan aft blade root to where the radius reaches a maximum. The localminimum location with radius r_(h) is found by proceeding axiallyforward from the bulge to where the radius stops decreasing. Theflowpath curve has ratio r_(b)/r_(h)>1.08. Furthermore, the axialdistance z_(b) between the bulge and the local minimum may conform tothe ratio z_(b)/r_(h)<2.41. Additionally, the flowpath curve may have alocation with radius r_(m) axially halfway between the bulge and thelocal minimum such that (r_(m)/r_(h)−1)/(r_(b)/r_(h)−1)>0.59. The aboveratios may be tailored to suit a predetermined cruise Mach number, M₀,as shown in EQS. 1, 2, and 3 presented sequentially below:

$\begin{matrix}{\frac{r_{b}}{r_{h}} = {{\left( {{A1} - 1} \right)\frac{M_{0} - 0.6}{0.19}} + 1}} & \left\lbrack {{EQ}.1} \right\rbrack\end{matrix}$ $\begin{matrix}{\frac{z_{b}}{r_{h}} = {B1\left( \frac{M_{0}}{0.79} \right)^{3}}} & \left\lbrack {{EQ}.2} \right\rbrack\end{matrix}$ $\begin{matrix}{\frac{{r_{m}/r_{h}} - 1}{{r_{b}/r_{h}} - 1} = {C1}} & \left\lbrack {{EQ}.3} \right\rbrack\end{matrix}$

In the above equations, 0.74<M₀<0.86, and constants A1, B1, and C1 haveranges 1.11<A1<1.31, 1.23<B1<1.63, and 0.59<C1<0.79. The aboverelationships for a flowpath curve corresponding to an aft blade rootmay apply to flowpath curves associated with multiple aft blade roots,or the flowpath curves associated with all aft blade roots.

According to the disclosure, an unducted propulsion system for asubsonic aircraft having a cruise Mach number, M₀, 0.74 or greater, forexample 0.74<M₀<0.86, has an axis of rotation, a forward blade assembly,an aft blade assembly, a forward housing, and an aft housing. Theforward and aft blade assemblies each include a plurality of blades,each blade having a root proximal to the axis of rotation and a tipdistal from the axis of rotation. A flowpath curve corresponds to theintersection of the forward housing external surface with a planecontaining the axis of rotation and the forward-most point of a forwardblade root. For the flowpath curve, axial direction, z, is parallel tothe axis of rotation, increasing in the forward or upstream direction.For the flowpath curve, radius, r, is distance from the axis ofrotation. At axial locations where the forward housing is rotating aboutthe axis of rotation (e.g., a spinner), radius, r, is an effectiveradius, i.e., the radius of a circle having the same cross-sectionalarea of the forward housing perpendicular to the axis of rotation.

The flowpath curve has a bulge and a local minimum. The bulge withradius r₁ is found by proceeding forward from the forward-most point ofthe forward blade root to where the radius reaches a maximum. The localminimum location with radius r₂ is found by proceeding aft from thebulge to where the radius stops decreasing within the axial extent ofthe forward blade root. The flowpath curve has ratio r₁/r₂>1.029.Furthermore, the axial distance z₁ between the bulge and local minimummay conform to the ratio z₁/r₂<1.522. Additionally, the forward housingmay have a forward-most point wherein the axial distance z₂ between thelocal minimum and the forward-most end of the flowpath curve may conformto the ratio z₂/r₂<4.115. The above ratios may be tailored to suit apredetermined cruise Mach number, M₀, as shown in EQS. 4, 5, and 6presented sequentially below:

$\begin{matrix}{\frac{r_{1}}{r_{2}} = {{\left( {{A2} - 1} \right)\frac{M_{0} - 0.6}{0.19}} + 1}} & \left\lbrack {{EQ}.4} \right\rbrack\end{matrix}$ $\begin{matrix}{\frac{z_{1}}{r_{2}} = {B2\left( \frac{M_{0}}{0.79} \right)^{3}}} & \left\lbrack {{EQ}.5} \right\rbrack\end{matrix}$ $\begin{matrix}{\frac{z_{2}}{r_{2}} = {C2\left( \frac{M_{0}}{0.79} \right)^{3}}} & \left\lbrack {{EQ}.6} \right\rbrack\end{matrix}$

Where, 0.74<M₀<0.86, 1.04<A2<1.14, 0.78<B2<1.18, and 2.19<C2<3.19.

Also according to the disclosure, an unducted propulsion system for asubsonic aircraft having a cruise Mach number, M₀, 0.74 or greater, forexample 0.74<M₀<0.86, includes a rotating element comprised of an axisof rotation, a forward blade assembly, and a forward housing. Theforward housing, or spinner, rotates with the forward blade assemblyabout the axis of rotation. The forward blade assembly includes aplurality of blades, each blade having a root proximal to the axis ofrotation and a tip distal from the axis of rotation. The axialdirection, z, for the spinner is parallel to the axis of rotation,increasing in the forward or upstream direction. The radius, r, of thespinner shape is the distance from the axis of rotation. Radialcoordinate, r, is an effective radius, i.e., a radius of a circle havingthe same cross-sectional area of the spinner perpendicular to the axisof rotation. The spinner has a bulge location with radius r₁ at amaximum radius forward of the forward blade assembly. Proceeding axiallyaft from the bulge, the spinner has a local minimum with radius r₂ wherethe radius stops decreasing within the axial extent of the forward bladeroots. The spinner is shaped such that ratio r₁/r₂>1.066. Furthermore,the axial distance z₁ between the bulge and local minimum may conform tothe ratio z₁/r₂<1.522. Additionally, the forward housing may have aforward-most point wherein the axial distance z₂ between the localminimum and the forward-most end of the flowpath curve may conform tothe ratio z₂/r₂<4.115. The above ratios may be tailored to suit apredetermined cruise Mach number, M₀, using EQS. 4, 5, and 6, where0.74<M₀<0.86, 1.09<A2<1.14, 0.78<B2<1.18, and 2.19<C2<3.19

These and other features, aspects and advantages of the presentdisclosure and/or embodiments will become better understood withreference to the following description and appended claims. Theaccompanying drawings, which are incorporated in and constitute a partof this specification illustrate embodiments of the present disclosureand, together with the description, serve to explain the principles ofthe present disclosure.

Here and throughout the specification and claims, range limitations arecombined and interchanged, such ranges are identified and include allthe sub-ranges contained therein unless context or language indicatesotherwise. For example, all ranges disclosed herein are inclusive of theendpoints, and the endpoints are independently combinable with eachother.

In the Figures which follow, like reference numerals are utilized torefer to like elements throughout the various embodiments depicted inthe Figures.

FIG. 1 shows an elevational cross-sectional view of an exemplaryunducted propulsion system 100. As is seen from FIG. 1 , the unductedpropulsion system 100 takes the form of an open rotor propulsion systemand has a rotating element 138 depicted as a propeller assembly whichincludes an array of blades 102 affixed to forward housing 106 andconfigured to rotate around an axis of rotation 120 of the unductedpropulsion system 100. The unducted propulsion system 100 also includesin the exemplary embodiment a non-rotating stationary element 142 whichincludes an array of blades 104, also known as vanes, disposed aroundaxis of rotation 120. These blades may be arranged such that they arenot all equidistant from the propeller. These blades are mounted to astationary frame and do not rotate relative to the central axis 120. Thenon-rotating stationary element 142 includes a stationary aft housing126. Forward housing 106 and aft housing 126 have external surfaces thatare three-dimensional. To explain the surface shaping guidance disclosedherein, parameters are defined along flowpath curves that correspond tointersecting the external surfaces with a plane that includes the axisof rotation. Accordingly, flowpath curve 105 corresponds to theintersection of forward housing 106 with an z-r plane that includes theaxis of rotation. Similarly, the flowpath curve 125 corresponds to theintersection of aft housing 126 with an z-r plane that includes the axisof rotation. For reference purposes, FIG. 1 also depicts a forwarddirection denoted with arrow 118.

As shown in FIG. 1 , the exemplary unducted propulsion system 100 alsoincludes a drive mechanism 128 which provides torque and power to therotating element 138 through a transmission (not shown). In variousembodiments, the drive mechanism 128, also known as an engine, may be agas turbine engine, an electric motor, an internal combustion engine, orany other suitable source of torque and power and may be located inproximity to the rotating element 138 or may be remotely located with asuitably configured transmission. Transmission transfers power andtorque from the drive mechanism 128 to the rotating element 138 and mayinclude one or more shafts, gearboxes, or other mechanical or fluiddrive systems. In FIG. 1 , drive mechanism 128 is depicted schematicallyas comprising a gas generator 130 and a power turbine 132. An example ofa turbomachine comprising a gas generator (e.g., compressor, combustor &high-speed turbine) and power turbine of a gas turbine engine is shownand described in US20210108597, hereby incorporated by reference in itsentirety for all purposes. Alternative configurations to the oneillustrated in FIG. 1 herein are depicted in U.S. Ser. No. 10/704,410,U.S. Pat. Nos. 5,190,441, 9,340,277, and U.S. Ser. No. 10/358,926,hereby each incorporated by reference in their entirety for allpurposes.

Airfoil blades 102 of rotating element 138 are sized, shaped, andconfigured to produce thrust by moving a working fluid such as air in adirection 144 as shown in FIG. 1 when the rotating element 138 isrotated in a given direction around the axis of rotation 120. In doingso, blades 102 impart a degree of swirl to the fluid as it travels inthe direction 144. Blades 104 of the stationary element are sized,shaped, and configured to decrease the swirl magnitude of the fluid, soas to increase the kinetic energy that generates thrust for a givenshaft power input to the rotating element. Each rotating blade 102 hasblade root 122 and blade tip 124. Each stationary blade 104 has a bladeroot 136 and blade tip 134. For both rotating blades 102 and stationaryblades 104, span is defined as the distance between root and tip.Stationary blades 104 may have a shorter span than rotating blades 102,for example, 50% of the span of blades 102, or may have longer span orthe same span as blades 102 as desired. In FIG. 1 , stationary blades104 are shown affixed to the housing 126 at their respective blade roots136. In some embodiments some or all stationary blades 104 may beaffixed to, or integrated with, the aircraft structure, such as a wing,pylon, or fuselage. The number of blades 104 of the stationary elementmay be fewer or greater than, or the same as, the number of blades 102of the rotating element and is typically greater than two, or greaterthan four. In some embodiments a ratio of the number of rotating blades102 to a number of stationary blades 104 is between 2:5 and 2:1. In someembodiments a difference between the number of rotating blades 102 to anumber of stationary blades 104 is between 2 and −2.

Blades 104 of the stationary element 142 may be positionedaerodynamically upstream of the rotating blades 102 to serve as counterswirl vanes, i.e., imparting a tangential velocity which is opposite tothe rotation direction of the rotating element 138. Alternatively, andas shown in FIG. 1 , stationary blades 104 may be positionedaerodynamically downstream of the rotating blades 102 to serve asde-swirl vanes, i.e., imparting a change in tangential velocity which iscounter to that of the rotating element 138. Any swirl remaining in theairstream downstream of the unducted propulsion system 100 equates to aloss in thrust producing kinetic energy.

It may be desirable that either or both of the sets of rotating blades102 and stationary blades 104 incorporate a pitch change mechanism suchthat the blades can be rotated with respect to an axis of pitch rotationeither independently or in conjunction with one another. Such pitchchange can be utilized to vary thrust and/or swirl effects under variousoperating conditions, including to provide a thrust reversing featurewhich may be useful in certain operating conditions such as upon landingan aircraft.

An inlet 127 is located axially between the blades 104 and blades 102.Alternatively, the inlet 127 may be located elsewhere, for example,forward of the blades 102. A ratio of a mass of air accelerated by therotating blades 102 and bypassing the inlet 127 to the mass of airaccelerated by the rotating blades and entering the engine core (notshown) via the inlet 127 is known as the bypass ratio. In someembodiments, a ratio of the sweep area of the blades (computed asπ×[(blade tip radius)²−(blade root radius)²) to the cross-sectional areaof the inlet (as measured in a z-r plane) is greater than 20:1 orgreater than 30:1, and less than 80:1.

It will be appreciated that the exemplary unducted propulsion system 100depicted in FIG. 1 is by way of example only. In other exemplaryembodiments, it may have other suitable configurations. For example,instead of being a forward rotating blade assembly and an aft stationaryblade assembly as shown, the two blade assemblies could becounter-rotating with respect to one another. As another example, theforward blade assembly could be stationary and the aft blade assemblycould be rotating. As another example, the unducted propulsion systemmay consist of only a rotating blade assembly, i.e., a propeller.

FIG. 2 is a perspective view of an exemplary gas turbine engine attachedto a wing of an aircraft in accordance with some aspects of the presentdisclosure. FIG. 2 depicts the unducted propulsion system 100 mounted toa wing 218 via a pylon 220 to facilitate mounting to, or accommodationof the airframe structure. Additionally, each of blades may not all beequally spaced from each other and/or at the same axial, z, location.These are examples of where a housing 126 may not be axisymmetric.

The unducted propulsion system 100 includes a turbomachine substantiallycontained within a forward housing or spinner 106 and aft housing 126.In some configurations, both the forward housing 106 and the aft housing126 include rotating hubs associated with rotating blades 102 and 104,respectively. In other configurations, one of the forward housing 106and the aft housing 126 are entirely rotating or include a rotatingstructure such as a rotating hub, while the other is a stationaryhousing associated with respective rotating and stationary blades. Insome embodiments, forward housing 106 may be considered a spinner andaft housing 126 may be considered a nacelle. Aft housing 126 may containa compressor, combustor, and turbine of a turbomachine, followed byengine exit 121.

In an illustrative non-limiting example as depicted in FIG. 2 , theunducted propulsion system 100 includes a rotating assembly (or rotor)that includes a forward housing 106 and airfoil-shaped assembly ofblades 102 (may also be referred to as a fan, rotor, or propeller)associated with the forward housing 106. In this example, the forwardhousing 106 is a spinner that rotates about an axis of rotation 120. Inother configurations the forward housing may not be rotating, as whenthe system is comprised of a stationary forward blade assembly androtating aft blade assembly. The unducted propulsion system 100 alsoincludes a stationary assembly that may include the engine inlet 127 andairfoil shaped stationary assembly of blades 104 associated with afthousing 126. In such a configuration, the housing 126 is non-rotating,as are the blades 104, about the axis 120 although the blades mayseparately articulate to modify a pitch, lean or sweep angle, e.g., viaa mechanism contained within housing 126. At least one of the functionsof the stationary assembly of blades 104 is to remove swirl from airstream leaving the rotor.

Aft housing 126 extends in an axial direction from engine inlet 127 tothe engine exit 121. The aft housing 126 contains the internal machinerythat produces torque for the assembly of blades 102 and defines asurface shaped to provide aerodynamic efficiency (reduce drag) for airpassing through blades 102 and 104 and proceeding downstream. The streamexhausted from the engine exit 121 produces some of the thrust thatpropels and/or advances an aircraft forward. Most of the thrust producedby an engine of the unducted propulsion system 100 comes fromaccelerated air that passes over the housing 126, or the air that passesthrough the blades 104 and bypasses the inlet 127. In some embodiments,the engine may additionally include a third stream (the first and secondstreams being the bypass and turbomachine core airstream defined by acompressor, combustor, and turbine).

For simplicity of illustration in FIG. 2 , forward housing 106 is shownas a continuous spinner. However, each housing may be comprised ofseparate parts with various mechanical components to allow variablepitch angle of the forward assembly of blades 102 and/or aft assembly ofblades 104. The axial extent of such specialized parts of each housingmay be approximately the same as a corresponding axial extent of theassembly of blades 102 and/or assembly of blades 104, or the axialextents of the housings may be shorter or longer (in axial extent) thanthe span of blades or respective axial extents of the blade assemblies.The dot-dashed line in FIG. 2 indicates an axis of rotation 120 for theblades 102. The dashed curves, 105 and 125, represent flowpath curvescorresponding to the intersection of housings 106 and 126, respectively,with a plane that includes axis of rotation 120. In the illustrativeexample in which forward housing 106 and the associated forward assemblyof blades 102 are rotating about axis of rotation 120, the shape of theflowpath curve may be defined by an effective radius vs. axial distanceparallel to the axis of rotation 120. However, in this example in whichthe aft housing 126 and the associated aft assembly of blades 104 do notrotate about axis of rotation 120, the flowpath curve shape of radiusvs. axal location depends on the orientation of the z-r plane about theaxis of rotation, i.e., curve could have a different shape for differentpositions of the plane that intersects aft housing 126.

Referring to FIGS. 1 and 2 , a flow restriction, known as blockage, forthe flow of air passing through the row of blades 102 and/or blades 104may be presented due to the assembly of blades 102 and/or the stationaryassembly of blades 104 having thickness. Thus, not only does the flow ofair accelerate through the row due to the generation of thrust, but theflow of air must accelerate further due to the blockage. At a highsubsonic cruise, such as a flight Mach number (M₀) greater than about0.74, these combined effects of the generated thrust and the blockagecan cause the axial component of the velocity of the flow of air throughthe row to approach the speed of sound (i.e., Mach number of 1), knownas choking, which can lead to high pressure loss within the blade 102 orblade 104 passage. Higher blade 102 or blade 104 counts, such as 8 to18, can make choking a major concern because increasing the countincreases the overall blockage of blade material for the air streambeing accelerated by the assembly of blades 102.

A strategy known as area ruling can reduce the Mach number in thepassages within the blades 102 or blades 104. To visualize area ruling,FIG. 3 shows a section view of unducted propulsion system 100. Flowpathcurve 125 corresponds to the intersection of the external surface ofhousing 126 of FIGS. 1 and 2 with a section plane that includes axis ofrotation 120 as well as the aft-most point of an aft blade root 136 of ablade 104 in the aft blade assembly. Thus, a point on the externalsurface of housing 126 is determined by choosing a blade root 136 andthe distance upstream or downstream parallel to the axis of rotation 120from the aft-most point of the blade root 136. By making an aft housingsurface concave 404, the Mach number within the passage of blades 104can be reduced. The aft housing concave region 404 corresponds to avalley and locates a local minimum radius of the surface of the housing.Referring to the corresponding flow of air through the blades 104 andover the housing, i.e., the flowpath curve 125, there is seen adesirable reduction in velocity due to the concave shape of the housingat the location of blades 104. To achieve this concave region 404, whichproduced a desired result (lowered Mach number at the blades 104 toavoid choking) the radial distance of curve 125 from the axis ofrotation 120 away from the blades 104 must increase, leading to convex406 curvature downstream and possibly convex curvature 402 upstream.Thus, not only may the housing 126 need to bulge outward to accommodateinternal components of the propulsion system, but it may also need tobulge outward to avoid choking in the passages between blades 104.

There may also be formed on the housing a convex portion 402 on thehousing surface upstream of the blades 104. Thus, the housing 126 maybulge outward to accommodate internal components of the unductedpropulsion system and bulge upstream for, e.g., accommodating componentsor the inlet 127. It may also be desirable to minimize the axial lengthof the unducted propulsion system. The goal of avoiding choking whilelimiting axial length may result in increasing the surface curvature ofthe housing 126, thereby causing local accelerations of the air alongthe flowpath curve 125, particularly at a convex portion. As describedbelow, curvature near a convex portion of the surface of the flowpathcurve 125 can present a challenge at high subsonic flight.

FIG. 4 illustrates the flow of air through the fan of an unductedpropulsion system 100 of FIG. 1 . Airspeed relative to the unductedpropulsion system 100 and far upstream 502 of the unducted propulsionsystem 100 has a velocity V₀ (e.g., corresponding to 0.74<M₀<0.86),which is the flight speed of the aircraft. Closer to the fan, theinfluence of the fan is to induce a higher velocity of air as the airenters the fan. As the air passes through the fan, the fan adds power tothe stream of air passing through it, to accelerate (i.e., furtherincrease the velocity) of the air that passes over the remainder of thepropulsion system. In an area far downstream 506 in the axial direction,the stream of air reaches an exhaust velocity, V_(e).

The stream of air that passes through the assembly of blades 104, fromfar upstream to far downstream, can be viewed as a tube of air (or fanstream tube) 508. The radial and axial extent of the fan stream tube 508(airflow of the slipstream) is indicated by the hashed region. An outerboundary 510 of the fan stream tube 508 intersects a radially outermostsection (or tip) 124 of the assembly of blades 102. An inner boundary514 of the fan stream tube 508 intersects the assembly of blades 104near the flowpath curve 125 and follows the shape of the flowpath curve125 immediately downstream of the assembly of blades 104. Because inthis illustrative example the engine inlet 127 ingests air from theinnermost radial region between blades 102 and 104, the fan stream tube508 excludes the portion of air passing through the blades 102 thatenters the engine inlet 127 and exhausts through the engine exit 121.The average axial velocity of air at any axial location within the fanstream tube 508 can be visualized by an annular cross-sectional area 516of the fan stream tube 508 at that location. Examples of an annularcross-sectional area 516 of the fan stream tube 508 are the far upstream502, a nacelle bulge 504, and the far downstream 506 for selectedlocations along the fan stream tube 508.

Because the mass flow rate of the air through any annular area withinthe stream tube 508 and downstream of the inlet 127 is the same and theair density is roughly constant throughout the fan stream tube 508, theaverage axial velocity of air is roughly inversely proportional to theannular area 516. Thus, far upstream 502, where the velocity of the fanstream tube 508 entering the fan has not yet increased due to the fan,the annular area 516 is largest. Far downstream 506, the fan stream tube508 includes energized air at a higher velocity relative to the velocityof air in the far upstream 502, so the annular area 516 is smaller thanin 502. The smallest annular area relative to the annular areas alongthe fan stream tube 508 occurs over the housing 126 near bulge 504. Atthe nacelle bulge 504, the air has been energized by the assembly ofblades 104, the radial distance from the axis of rotation is at amaximum radius, and the annular area 516 is the smallest relative to theother mentioned annular areas of the flowpath curve 125. Thus, theaverage axial velocity of the flow of air over the housing 126 (definingthe surface of the flowpath curve 125) is high and attributable to abulge in the flowpath curve 125.

The problem due to the high average axial velocity of the flow of airover the nacelle is further explained in FIG. 5 , which depicts theeffects when air flows without friction from left to right over a wavysolid surface 604. Streamlines 602 indicate the path of fluid particlesstarting at various distances from the surface 604. A concave surface606, or valley, increases the static pressure and reduces the velocityof the air. In contrast, a convex surface 608, or peak, decreases thestatic pressure and increases the velocity of the air. Thus, for flowover housing 126 of FIG. 1 , the change in static pressure and theaccompanying opposite change in velocity of air is governed in largepart by the curvature associated with the housing 126.

Curvature of a surface can be expressed in terms of a correspondingradius of curvature. For example, at any point along the surface 604,one can define the radius of curvature, r_(c), and a center of curvature610. To illustrate, two radii of curvatures 612, 614 and theircorresponding centers (shown as a “+”) 610 corresponding to two surfacelocations are shown in FIG. 5 . At a distance to the left of the peak ofconvex surface 608, the curvature is low, which corresponds to a larger_(c) 614. Nearer to the peak of convex surface 608, the curvature ishigh, which corresponds to a small r_(c) 612. Locations within theconcave surface 606 also have low and high curvature. However, forpoints within concave surface 606, the center of curvature is locatedabove curve 604 with the radius of curvature pointing towards thesurface 604.

As explained above, the flow of air over the housing 126 can have ahigher average velocity than the fan flow downstream of the engine,V_(e) and this effect can present a problem for a high subsonic flight.In particular, as the speed of air over the flowpath surface 125approaches the speed of sound, or Mach=1.0, the drag begins to increasesharply. In general, friction drag increases roughly in proportion tothe square of the air velocity. However, as the Mach number increases, alarger contributor to the increased in drag comes from wave drag. A wavedrag is a drag resulting from shock waves that form as the flow of airnear the housing surface 126 becomes supersonic (e.g., Mach>1.0).

The above explanation illustrates three factors that contribute to highdrag. A first factor is high cruise flight Mach, M₀, for example0.74<M₀<0.86. A second factor is high non-dimensional cruise fan netthrust based on fan annular area and flight speed. The same accelerationof the air stream by the fan that produces thrust also tends to increasethe drag force on housing 126 (e.g., nacelle). Expressing thrustnon-dimensionally in a way that accounts for flight speed, ambientconditions, and fan annular area yields a thrust parameter

$\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}},$where F_(net) is cruise fan net thrust, ρ₀ is ambient air density, V₀ iscruise flight velocity, and A_(an) is fan stream tube cross-sectionalarea at the fan inlet. Fan annular area, A_(an), is computed using amaximum radius as the tip radius of the forward-most rotor blades and aminimum radius as the minimum radius of the fan stream tube entering thefan. A third factor is a large ratio of maximum radius of housing 126relative to the local minimum radius associated with the aft blade root136 combined with a relatively small ratio of axial length between thelocal minimum radius and the maximum radius for housing 126 to the localminimum radius associated with the aft blade root 136.

A solution to the problem presented by wave drag at a high subsonicflight (e.g., 0.74<M₀<0.86) is to design a shape of a flowpath curve 125on the housing 126 based on an unconventional surface curvaturestrategy. FIG. 6 shows a schematic illustration of three surfacelocations 702, 704, and 706 on the flowpath curve 125. As for FIGS. 3and 4 , flowpath curve 125 corresponds to the intersection of theexternal surface of housing 126 shown in FIGS. 1 and 2 with a plane thatincludes axis of rotation 120 and the aft-most point of an aft bladeroot 136 in the aft blade assembly. Thus, curve 125 corresponds toproceeding along the surface of housing 126 axially forward and aft fromthe aft-most point on an aft blade root 136. If the aft blade used todefine the flowpath curve 125 has variable orientation, such as actuatedby a pitch change mechanism, the most relevant aft blade orientation forlocating the flowpath curve is when the aft-most point of the aft bladeroot 136 is furthest aft. In case the aft-most point of the aft bladeroot 136 is not attached to housing 126, e.g., there is a clearance gapbetween the aft blade root 136 and the housing 126 to allow for pitchchange or the aft blade 104 is attached to the airframe and suspendedover the housing 126, then curve 125 goes through the nearest point onthe surface of housing 126 to the aft-most point of the aft blade root136. Each surface location along flowpath curve 125 can be defined basedon an (z, r) coordinate system 708 in which the z-axis is the axis ofrotation 120 and r is the distance from the axis of rotation 120.

Flowpath curve 125 has a bulge, or maximum radius location 704corresponding to (z_(b), r_(b)) of the (z, r) coordinate system 708 andhas a maximum radius r_(b). Flowpath curve 125 has a local minimumlocation 702 forward of the bulge location at 704 corresponding to (0,r_(h)) of the (z, r) coordinate system 708 and has a radius r_(h).Surface locations at (0, r_(h)) and (z_(b), r_(b)) determine the axialand radial extent of a segment of the flowpath curve 125 where the shapeis designed as described herein to solve the problem of high wave dragfor high subsonic flight.

Flowpath curve 125 has a third location 706 corresponding to (z_(b)/2,r_(m)) in the (z, r) coordinate system 708 and has an axial distancehalfway between the first surface location 702 and the second surfacelocation 704. For fixed endpoints 702 and 704 for the segment of curve125, specifying location of 706 has a strong effect on the distributionof curvature. Radii r_(h) 113, r_(m) 117, r_(b) 111, and axial distancez_(b) 115 are also shown in FIG. 1 .

For high subsonic cruise, achieving low drag without unwanted lengthincrease for housing 126 at high subsonic cruise Mach number, i.e.M₀>0.74, depends on appropriate positioning ofpoints/endpoints/locations 702, 704, and 706. For example, forsufficient bulge to suppress the Mach number within the aft bladeassembly, limited length to avoid excessive friction drag and weight,and limited convex curvature approaching bulge, it may be desirable forr_(b)/r_(h)>1.081, z_(b)/r_(h)<2.103, and(r_(m)/r_(h)−1)/(r_(b)/r_(h)−1)>0.59. Better results may be obtainedwith somewhat larger radius increases and shorter axial distance suchthat r_(b)/r_(h)>1.118, z_(b)/r_(h)<1.974, and(r_(m)/r_(h)−1)/(r_(b)/r_(h)−1)>0.64. Also, it may be beneficial toimpose an upper limit on the bulge such that r_(b)/r_(h)<1.424.

Additionally, the above ratios may be tailored to suit a pre-determinedcruise flight Mach number, M₀, with constants A1, B1, and C1, as shownin EQs 1, 2, and 3

$\begin{matrix}{\frac{r_{b}}{r_{h}} = {{\left( {{A1} - 1} \right)\frac{M_{0} - 0.6}{0.19}} + 1}} & {{EQ}.1}\end{matrix}$ $\begin{matrix}{\frac{z_{b}}{r_{h}} = {B1\left( {M_{0}/0.79} \right)^{3}}} & {{EQ}.2}\end{matrix}$ $\begin{matrix}{\frac{{r_{m}/r_{h}} - 1}{{r_{b}/r_{h}} - 1} = {C1}} & {{EQ}.3}\end{matrix}$

Where M₀>0.74, A1>1.11, B1<1.63, and C1>0.59. Additional limits on eachparameter may yield a more optimum configuration, e.g., 0.74<M₀<0.86,1.11<A1<1.31, 1.23<B1<1.63, and 0.59<C1<0.79. An example of furtherconstraints on the constants used to configure aft housing 126 include1.16<A1<1.31, 1.23<B1<1.53, and 0.64<C1<0.79. As another example ofconstraints on the constants, 1.16<A1<1.26, 1.33<B1<1.53, and0.64<C1<0.74.

Table 1 provides examples for the ratio of the bulge radius (r_(b)) 111to the local minimum radius (r_(h)) 113 where 1.11<A1<1.31 (in bold) and0.74<M₀<0.86.

TABLE 1 A1 values & r_(b)/r_(h) M₀ 1.11 1.16 1.21 1.26 1.31 0.74 1.0811.118 1.155 1.192 1.228 0.79 1.110 1.160 1.210 1.260 1.310 0.84 1.1391.202 1.265 1.328 1.392 0.86 1.151 1.219 1.287 1.356 1.424

Table 2 provides examples for the ratio of the axial distance 115between the local minimum and the bulge location and the local minimumradius (r_(h)) 113 where 1.23<B1<1.63 (in bold) and 0.74<M₀<0.86.

TABLE 2 B1 values & z_(b)/r_(h) M₀ 1.23 1.33 1.43 1.53 1.63 0.74 1.0111.093 1.175 1.257 1.340 0.79 1.230 1.330 1.430 1.530 1.630 0.84 1.4791.599 1.719 1.839 1.959 0.86 1.587 1.716 1.845 1.974 2.103

Table 3 provides examples for the ratio (r_(m)/r_(h)−1)/(r_(b)/r_(h)−1)where 0.59<C1<0.79 (in bold) and 0.74<M₀<0.86.

TABLE 3 C1 values & (r_(m) − r_(h))/(r_(b) − r_(h)) M₀ 0.59 0.64 0.690.74 0.79 0.74 0.59 0.64 0.69 0.74 0.79 0.79 0.59 0.64 0.69 0.74 0.790.84 0.59 0.64 0.69 0.74 0.79 0.86 0.59 0.64 0.69 0.74 0.79

In addition to applying to a range of cruise flight Mach number, M₀, theabove constraints on curve 125 may be particularly beneficial for arange of a dimensionless cruise fan net thrust parameter normalized byambient density, cruise flight speed squared, and fan stream tubeannular area at fan inlet,

$\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}}.$

In the above thrust parameter, F_(net) is cruise fan net thrust, ρ₀ isambient air density, V₀ is cruise flight velocity, and A_(an) is annularcross-sectional area perpendicular to the axis of rotation of the fanstream tube entering the fan. For the illustrative example shown in FIG.1 , the annular area would be computed using r_(t) 101, the radialdistance from the axis of rotation 120 to a tip end of a blade 102 inthe forward blade assembly, and the minimum radius of the fan streamtube at the same axial location. For the example of FIG. 1 in which theengine inlet stream occupies a portion of the forward blade assemblyannular area, a method to estimate the minimum radius of the fan streamtube would be used by those skilled in the art, using parameters such asfan thrust, engine inlet flow, and flight conditions. The thrustparameter may be, greater than or equal to 0.060, (e.g., greater than0.080, or greater than 0.084).

The unconventional surface curvature strategy described above to solvethe problem presented by the wave drag for a sustained high subsonicflight (e.g., 0.74<M₀<0.86) is applicable for the unducted propulsionsystems as described herein. In some configurations, the unconventionalsurface curvature strategy may be applicable to an unducted propulsionsystem having no engine inlet (inlet 127 omitted); for example, a rotornot driven by an air-breathing engine, but driven by another type ofmachine, such as an electric motor. FIG. 7 depicts a graph 800 of threeexemplary flowpath curves 125 that may be used to define the surface ofhousing 126 shown in FIGS. 1 and 2 . The flowpath curves 125 proximateaft housing 126 and between the surface locations at (0, r_(h)) and(z_(b), r_(b)) of the (z, r) coordinate system 708 as shown in FIG. 6 .

To explain how the points 702, 704, and 706 in FIG. 6 define the shapeof the aft housing 126 to reduce drag at high speed flight, threeexemplary flowpath curves 125 between points 702 and points 704 areplotted with z and r nondimensionalized by the local minimum radiusr_(h) in graph 800 of FIG. 7 . To facilitate comparison, the threecurves conform to EQS. 1, 2, and 3 with M₀=0.79, A1=1.21, and B1=1.43,with differences only in parameter C1. Flowpath curve 802 corresponds toC1=0.50 and is described by a cubic polynomial shape, labeled “cubic”.Flowpath curve 802 gives a smooth variation in curvature relative tocurves 804 and 806. Flowpath curve 804 corresponds to C1=0.61 and islabeled “ex1”. Flowpath curve 806 corresponds to C1=0.69 and is labeled“ex2”. Over more than the first one-third of its length, flowpath curve804, designated “ex1,” has a more rapid increase radius with axialdistance than curve 802. Flowpath curve 806, designated “ex2,” also hasa more rapid increase in radius than curve 802, but has less variationin radius near the peak radius of the housing (the flowpath curve 125location having the maximum radius r_(b)) than either curve 802 or 804.FIG. 8 shows a graph 900 the first derivatives of r with respect to zfor the curves in FIG. 7 . All curves begin and end with firstderivative of zero because the ends are at the local minimum and maximumradii. Plots 902, 904 and 906 correspond to the first derivatives of thecubic, ex1, and ex2 curves in FIG. 7 .

FIG. 9 shows a graph 1000 of the second derivatives of r with respect toz for the three curves in FIG. 7 . Second derivative indicates curvatureand a positive second derivative indicates concave curvature while anegative second derivative indicates convex curvature. The absolutevalue of the second derivative indicates the magnitude of curvature.Curves 1002, 1004 and 1006 are the second derivatives for the flowpathcurves “cubic”, “ex1” and “ex2”, respectively, shown in FIG. 7 . Thecubic polynomial flowpath curve has the smoothest curvature variation(linear with axial distance). Flowpath curve “ex1” also has a monotonicvariation in curvature, however, its curvature 1004 starts higher nearthe aft blade root 136 and decreases continuously towards the maximumradius. “Front-loading” the curvature in this way results in a lowermagnitude of convex curvature at the maximum radius than curvature 1002.Flowpath curve “ex2” has larger variations in third curvature 1006 thatachieve suppression of the Mach number within the passages of blades 104and avoids high convex curvature immediately upstream of the maximumradius. Because curve “ex2” has relatively low convex curvature wherethe combined effects of fan stream tube acceleration and flowpath curveradius increase may otherwise lead to excessive Mach number, the thirdcurvature 1006 (“ex2”) may be preferred.

As previously discussed, the speed at which an aircraft can fly islimited by numerous factors. With respect to propeller-driven aircraft,the propeller plays an important role in the speed at which the aircraftcan fly. At a high level, the larger the propeller and/or the greaterthe number of blades the propeller has, the faster the aircraft can fly.Unfortunately, while speed is often directly proportional to both thesize of the propeller and the number of blades, so is weight, and largesize creates problems for propulsion system installation andfeasibility. For example, as the size of the propeller and/or the numberof blades increases, the weight of the propeller generally increases,and larger propellers can be difficult to accommodate while maintainingground or fuselage clearances for a select airframe configuration. Also,at high subsonic flight speeds, a larger number of blades increases theflow-area blockage in the propeller blade row, which is problematicgiven the transonic flow around the blades. In particular, too muchblockage decreases propeller efficiency and range of operability.Accordingly, creating an acceptable aircraft capable of flying at highersustained speeds (e.g., cruise speeds) requires more than increasing thesize of the propeller and/or increasing the number of propeller blades.

FIG. 10 shows the same cross-sectional view as in FIG. 1 , but withannotations made to the forward part of the unducted propulsion system100, in particular the rotating element 138, which comprises a forwardhousing depicted as spinner 106 and the plurality of blades 102. Theblades 102 have blade roots 122 and blade tips 124. The blades 102 areaffixed to the spinner 106 at the blade roots 122. The rotating element138 can have any suitable number of blades 102. For example, in oneembodiment, the rotating element 138 includes between 8 and 18 blades.The spinner 106 and the blades 102, as part of the rotating element 138,rotate about the axis of rotation 120. The spinner 106 has aforward-most point/end/location 108, relative to an arrow 118 indicatinga direction of travel of the unducted propulsion system 100 and thus theaircraft.

The forward housing 106 is shaped such that it has a varying radiusalong its axial length and its shape is viewed along a flowpath curve105 formed by the intersection of the spinner surface with a plane thatincludes the axis of rotation 120 and the forward-most point of aforward blade root 122. As stated previously, a flowpath curve isdefined by the effective radius at axial locations in which the housingis rotating. Thus, in the illustrative example of FIG. 10 , the forwardblade root 122 chosen to construct the plane does not affect theflowpath curve 105. However, in some embodiments, the forward housing106 may be stationary. Thus, the convention of specifying a forwardblade root 122 to define the plane, and thus, flowpath curve 105,applies in other embodiments as this facilitates defining the curve forembodiments in which forward housing 106 is stationary. Flowpath curve105 for spinner 106 has a bulge location at the axial location where theradius reaches a maximum axially forward of the forward-most point offorward blade root 122 of blade 102, determining the first radius 110(denoted “r₁” in FIG. 10 ). Flowpath curve 105 for spinner 106 has alocal minimum location where, proceeding axially aft from the bulge, theradius reaches the local minimum proximal to the blades 102, determiningthe second radius 112 (denoted “r₂” in FIG. 10 ). Thus, the axiallocation of the first radius 110 is located forward of the axiallocation of the second radius 112 (i.e., between the axial location ofthe second radius 112 and the forward-most location 108 of the spinner106). Span of blade 102 is defined as the distance between blade root122 and blade tip 124. In one embodiment, the blades 102 have a maximumaxial distance/width 140 near mid span (i.e., 50% of the blade heightfrom the blade root to blade tip). In one embodiment, the blades 102 areaffixed to the spinner 106 such that, when oriented or configured forcruise operation, the forward-most point of the blade roots 122 isproximal to the local minimum having second radius 112, and such that 0to 40 percent of the maximum width 140 is located forward of theforward-most point of the blade roots 122. In another embodiment, 20 to40 percent of the maximum width 140 is located forward of theforward-most point of the blade roots 122.

In one embodiment, the first radius 110 is greater than the secondradius 112 and thus defines a bulge of the spinner 106, the location onthe spinner proceeding axially forward from the forward blade root 122where the radius reaches a maximum. A first distance 114 (denoted by“z₁” in FIG. 10 ) is defined between the bulge having first radius 110and the local minimum having second radius 112. A second distance(denoted by “z₂” in FIG. 10 ) is defined between the forward-mostlocation 108 of the spinner 106 and the local minimum having secondradius 112. The various parameters (i.e., the first radius 110, thesecond radius 112, the first distance 114, and the second distance 116)may be specified based on a predetermined speed of the aircraft. Thatis, suitable values for the various parameters are dependent upon apredetermined range of speeds of the aircraft. In some embodiments, thepredetermined speed of the aircraft is based on a desired airspeed forthe aircraft. For example, the predetermined speed for the aircraft canbe a speed, or range of speeds, at which the aircraft is designed tooperate while cruising. The predetermined speed for the aircraft can beany suitable value(s) and can range, for example, between Mach 0.74 and0.86 also referred to herein as a high subsonic cruise speed. Though theexample predetermined speed range of the aircraft is given as betweenMach 0.74 and Mach 0.86, it should be noted that the range can begreater, or smaller, than the range provided and have higher and/orlower maximums and minimums. For example, predetermined flight Machnumber can be between 0.78 and 0.84.

At a high level, the size of the bulge (i.e., the ratio of the firstradius 110 to the second radius 112) beneficial for low pressure loss onthe spinner and within the assembly of blades 102 increases as thepredetermined speed of the aircraft increases. Put simply, the largerthe bulge, the lower the flow velocities through the row of blades 102for a particular flight speed. However, as the size of the bulgeincreases, the required length of the spinner 106 increases, increasingthe weight of the rotating element 138. Accordingly, the size of thebulge is dictated by a number of factors based on the predeterminedspeed of the aircraft. Additionally, the minimal dimension of the secondradius 112 is typically dictated by the equipment needed for therotating element 138, such as blade retention hardware, pitch changemechanisms, counterweight systems, gearbox, gearbox cooling systems,lubrication systems, bearings, and drive shaft.

In one embodiment comprised of a forward assembly of blades and an aftassembly of blades, the non-dimensional bulge radius is r₁/r₂>1.029. Inother embodiments, the size of the bulge is described by a ratio of thefirst radius 110 and the second radius 112 and defined by EQ. 4:

${\frac{r_{1}}{r_{2}} = {{\left( {{A2} - 1} \right)\frac{M_{0} - 0.6}{0.19}} + 1}},$

Where r₁ is the first radius 110, r₂ is the second radius 112 associatedwith housing 106, M₀ is Mach number for sustained high speed flight,such as cruise, of the aircraft, and A2 is a constant. In oneembodiment, the value of A2 is within the range from 1.04 to 1.14. Ascan be seen by EQ. 4, the size of the bulge (i.e., the ratio of thefirst radius 110 to the second radius 112) increases as thepredetermined speed of the aircraft increased for each value of A2within the range. Specifically, for a minimum value of A2=1.04, theratio of the first radius 110 to the second radius 112 is 1.029 forM₀=0.74, 1.040 for M₀=0.79, 1.051 for M₀=0.84, and 1.055 for M₀=0.86.For a maximum value A2=1.14, the ratio of the first radius 110 to thesecond radius 112 is 1.103 for M₀=0.74, 1.140 for M₀=0.79, 1.177 forM₀=0.84, and 1.192 for M₀=0.86. Table 4 provides examples for the ratioof the first radius 110 to the second radius 112 where 1.04<A2<1.14 (inbold) and 0.74<M₀<0.86.

TABLE 4 A2 values & r₁/r₂ M₀ 1.04 1.06 1.09 1.12 1.14 0.74 1.029 1.0441.066 1.088 1.103 0.79 1.040 1.060 1.090 1.120 1.140 0.84 1.051 1.0761.114 1.152 1.177 0.86 1.055 1.082 1.123 1.164 1.192

As previously discussed, the geometry of the spinner 106 can also bedescribed based on the first distance 114 (i.e., the axial distancebetween the bulge having first radius 110 and the local minimum havingsecond radius 112). In one embodiment comprised of a forward assembly ofblades and an aft assembly of blades, the non-dimensional axial distancez₁/r₂<1.522. In another embodiment, the first distance 114 is describedin terms of a ratio of the first distance 114 to the second radius 112and defined by EQ. 5:

$\frac{z_{1}}{r_{2}} = {B2\left( \frac{M_{0}}{{0.7}9} \right)^{3}}$

where z₁ is the first distance 114, r₂ is the second radius 112, M₀ isMach number for sustained high speed flight, such as cruise, of theaircraft, and B2 is a value. In one embodiment, the value of B2 iswithin the range from 0.78 to 1.18. As can be seen by EQ. 5, the firstdistance 114 increases as the predetermined speed of the aircraftincreases for each value of B2 within the range. Put simply, the lengthof the spinner 106 increases as the predetermined speed of the aircraftincreases. Specifically, for a minimum value of B2=0.78, the ratio ofthe first distance 114 to the second radius 112 is 0.641 for M₀=0.74,0.780 for M₀=0.79, and 0.938 for M₀=0.84, and 1.006 for M₀=0.86. For amaximum value B2=1.18, the ratio of the first distance 114 to the secondradius 112 is 0.970 for M₀=0.74, 1.180 for M₀=0.79, 1.419 for M₀=0.84,and 1.522 for M₀=0.86. Table 5 provides examples for the ratio of thefirst distance 114 to the second radius 112 where 0.78<B2<1.18 (in bold)and 0.74<M₀<0.86.

TABLE 5 B2 values & z₁/r₂ M₀ 0.78 0.88 0.98 1.08 1.18 0.74 0.641 0.7230.805 0.888 0.970 0.79 0.780 0.880 0.980 1.080 1.180 0.84 0.938 1.0581.178 1.298 1.419 0.86 1.006 1.135 1.264 1.393 1.522

As previously discussed, the geometry of the spinner 106 can also bedescribed based on a second distance 116 (i.e., a distance between theforward-most location 108 of the spinner 106 and the local minimumhaving second radius 112). In one embodiment comprised of a forwardassembly of blades and an aft assembly of blades, the non-dimensionalaxial distance z₂/r₂<4.115. In one embodiment, the second distance 116is described in terms of a ratio between the second distance 116 and thesecond radius 112 and defined by EQ. 6:

$\frac{z_{2}}{r_{2}} = {C2\left( \frac{M_{0}}{{0.7}9} \right)^{3}}$

where z₂ is the second distance 116, r₂ is the second radius 112, M₀ isMach number for sustained high speed flight, such as cruise, of theaircraft, and C2 is a value. In one embodiment, the value of C2 iswithin the range from 2.19 to 3.19. As can be seen by EQ. 6, the seconddistance 116 increases as the predetermined speed of the aircraftincreases. Put simply, the length of the spinner 106 increases as thepredetermined speed of the aircraft increases. Specifically, for aminimum value of C2=2.19, the ratio of the second distance 116 to thesecond radius 112 is 1.800 for M₀=0.74, 2.190 for M₀=0.79, 2.633 forM₀=0.84, and 2.825 for M₀=0.86. For a maximum value C2=3.19, the ratioof the second distance 116 to the second radius 112 is 2.622 forM₀=0.74, 3.190 for M₀=0.79, 3.835 for M₀=0.84, and 4.115 for M₀=0.86.Table 6 provides examples for the ratio of the second distance 216 tothe second radius 112 where 2.19<C2<3.19 (in bold) and 0.74<M₀<0.86.

TABLE 6 C2 values & Z₂/r₂ M₀ 2.19 2.39 2.69 2.99 3.19 0.74 1.800 1.9642.211 2.457 2.622 0.79 2.190 2.390 2.690 2.990 3.190 0.84 2.633 2.8733.234 3.594 3.835 0.86 2.825 3.083 3.470 3.857 4.115

While the discussion of FIG. 10 describes an unducted propulsion systemfor propelling an aircraft consistent with the teachings herein, thediscussion of FIG. 11 provides additional detail regarding a plot ofvalues for geometries of a spinner for such an unducted propulsionsystem.

FIG. 11 is a chart 200 depicting external flowpath shapes for a spinnerof an unducted propulsion system, according to some embodiments. TheY-Axis 204 represents the spinner radius normalized by the second radius112, r/r₂, the second radius being at the local minimum within the axialextent of blades 102 nearest to the bulge having first radius 110. TheX-Axis 202 represents the axial distance from the axial location of thesecond radius 112 (i.e., the local minimum) normalized by the secondradius 112, z/r₂.

The chart 200 illustrates the forward housing or spinner 106 shape fordifferent cruise Mach numbers, M₀. Specifically, the chart 200 includesa first plot 206, a second plot 208, a third plot 210, and a fourth plot212. Each of the first plot 206, the second plot 208, the third plots210, and the fourth plot 212 arise from the same values of A2=1.09,B2=0.98, and C2=2.69, but for a different cruise Mach number, M₀. Thefirst plot 206 corresponds to M₀=0.70, the second plot 208 correspondsto M₀=0.74, the third plot 210 corresponds to M₀=0.79, and the fourthplot 212 corresponds to M₀=0.84. As can be seen from the chart 200,which depicts approximate shapes and relative sizes of the spinners, theratio of the first radius to the second radius and the ratio of thesecond distance to the second ratio increase as the predetermined speedincreases.

In addition to specifying the forward housing dimension ratios, furtherconstraints on the shape of the flowpath curve are described herein. Asuperellipse equation below may provide a suitable distribution ofcurvature along the flowpath curve 105 to avoid excessive Mach numberalong the portion of the forward housing forward of the bulge. Inspecifying the shape of the spinner using the obtained r₁, z₁, and z₂, asuperellipse expression provides optional bounds on the flowpath curve105 forward of the bulge. EQ. 7 for a superellipse relating the axialcoordinate, z, to the radius, r, is given below.

${{\left( \frac{z - z_{1}}{z_{2} - z_{1}} \right)^{p} + \left( \frac{r}{r_{1}} \right)^{q}} = {1{or}}},{equivalently},{{\left( \frac{{z/r_{2}} - {z_{1}/r_{2}}}{{z_{2}/r_{2}} - {z_{1}/r_{2}}} \right)^{p} + \left( \frac{r/r_{2}}{r_{1}/r_{2}} \right)^{q}} = 1}$

In EQ. 7, exponents p and q define a shape of a curve forward of thebulge for ratios r₁/r₂, z₁/r₂, and z₂/r₂ determined via EQS. 4, 5, and 6as described above. FIG. 12 provides chart 1200 similar to chart 200 inFIG. 11 . Accordingly, X-Axis 202 and Y-Axis 204, and curve 210 in chart1200 are the same as in chart 200. Curves 1208 and 1212 conform to thesame ratios determined by EQS. 4, 5, and 6 as curve 210. However, curves1208 and 1212 that bound the range of suitable points for flowpath curve105 are determined by using EQ. 7 via values of exponents p and q. Curve1208, with exponents p=1.5 and q=2.0, forms a lower bound on suitablepoints for flowpath curve 105 forward of the bulge. Curve 1212, withexponents p=3.0 and q=3.5 forms an upper bound of suitable points forflowpath curve 105 forward of the bulge. Thus, within the axial rangefrom the bulge to the forward-most end 108 of the forward housing 106,EQ. 7 with ranges on exponents p and q provides a band, or range, ofpoints to define the shape of forward housing 106. Curve 210 agreesclosely with EQ. 7 using exponents p=2.0 and q=3.0. Thus, options forthe lower bound conform to exponent ranges 1.5<p<2.0 and 2.0<q<3.0 whileoptions for the upper bound conform to exponents ranges 2.0<p<3.0 and3.0<q<3.5. At least for some cruise flight Mach numbers, M₀, for example0.79 depicted as plot 210, a low loss flowpath curve can be obtainedwithin a more limited bounds such that the lower constraints on theflowpath curve are in the ranges 1.7<p<2.0 and 2.5<q<3.0 while the upperconstraints on the flowpath curve are in the ranges 2.0<p<2.5 and3.0<q<3.3.

In some configurations, the above spinner shape parameters may beparticularly beneficial for a range of a dimensionless cruise fan netthrust parameter. The thrust parameter is the same as defined earlier:

$\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}}.$The thrust parameter may be, greater than or equal to 0.060, (e.g.,greater than 0.080, or is greater than 0.084).

It should be recognized that the forward housing 106 or spinner need notbe axisymmetric about the axis of rotation for the propeller. Forexample, at the axial location of the second radius proximal to theplurality of blades, the distance of the spinner or hub surface may varyin the circumferential direction to accommodate blade attachment orvariable pitch mechanisms. As stated previously, for axial locationsalong forward housing 106 that are rotating about the axis of rotation120, as the case with a spinner, the radius, such as the second radius,is defined as an “effective” radius of a circle having the samecross-sectional area of the spinner normal to the axis of rotation.Thus, the term “radius” used in the description and claims refers to theradius of a circle having the cross-sectional area of the spinner atthat axial location. However, for a forward housing 106 that isstationary, as could be the case for an unducted propulsion system inwhich the forward blade assembly is stationary and the aft bladeassembly is rotating, the flowpath curve 105 corresponds to theintersection of the forward housing with a plane that includes the axisof rotation and the forward-most point of the forward blade root 122. Ifthe forward blade 102 has variable pitch, then the forward-most pointcorresponds to the blade orientation that positions the forward-mostpoint in its most forward position, likely approximately to the cruiseor design point condition. In this case, the flowpath curve 105disclosed herein may correspond to one of the blade roots 122, more thanone blade root, or all the blade roots. In case the forward-most pointof the forward blade root 122 is not attached to the forward housing106, e.g., there is a clearance gap between the forward blade root 122and the forward housing 106 to allow for pitch change or the forwardblade 102 is attached to the airframe and suspended over the forwardhousing 106, then curve 105 goes through the nearest point on thesurface of the forward housing 106 to the forward-most point of theforward blade root 122.

In some embodiments, a rotating element for an unducted propulsionsystem for propelling an aircraft comprises a plurality of bladesaffixed to a spinner, wherein the spinner is configured to rotate aboutan axis of rotation, wherein the spinner includes a first radius and asecond radius, wherein the second radius is proximal to the plurality ofblades and the first radius is forward from the second radius, wherein aratio of the first radius to the second radius ranges from 1.029 to1.192, and wherein the aircraft is configured to travel at apredetermined speed.

In some embodiments, a rotating element for an unducted propulsionsystem for propelling an aircraft comprises a plurality of bladesaffixed to a spinner, wherein the spinner is configured to rotate aboutan axis of rotation, wherein the flowpath curve on the spinner includesa first radius and a second radius, wherein the first radius is at thebulge or maximum radius forward of the associated plurality of blades,wherein the second radius is at the local minimum aft of the bulge,wherein a first distance is defined between the axial locations of thebulge and the local minimum, and wherein a second distance is definedbetween a forward-most end of the spinner and the axial location of thelocal minimum, wherein a ratio of the first radius to the second radiusranges from 1.029 to 1.192, wherein a ratio of the first distance to thesecond radius ranges from 0.641 to 1.522, wherein a ratio of the seconddistance to the second radius ranges from 1.800 to 4.115, wherein theaircraft is configured to travel at a predetermined speed. FIG. 13 is aflow chart of a method 1300 of operating an unducted propulsion systemfor propelling an aircraft. The unducted propulsion system includes aspinner and a plurality of blades affixed to the spinner. The methodincludes the steps of rotating 1302 the spinner about an axis ofrotation and operating 1304 the aircraft at a predetermined speed ofgreater than or equal to Mach 0.74. The spinner may be configured asdescribed herein with respect to FIGS. 1 and 2 . For example, thespinner may include a first radius and a second radius, wherein thesecond radius is proximal to the plurality of blades and the firstradius is forward from the second radius, wherein a ratio of the firstradius to the second radius is greater than 1.029. Further, the ratio ofthe first radius to the second radius may be defined by EQ 4:

${\frac{r_{1}}{r_{2}} = {{\left( {{A2} - 1} \right)\frac{M_{0} - {{0.6}0}}{{0.1}9}} + 1}},$

wherein r₁ is the first radius, r₂ is the second radius, M₀ correspondsto a predetermined sustained high speed of the aircraft (such ascruise), and A2 is a value that ranges from 1.04 to 1.14.

Also, a first distance is defined between the axial locationcorresponding to the first radius and the axial location correspondingto the second radius, and the ratio of the first distance to the secondradius is less than 1.522. Further, the ratio of the first distance tothe second radius may be defined by EQ. 5:

$\frac{z_{1}}{r_{2}} = {B2\left( \frac{M_{0}}{{0.7}9} \right)^{3}}$

wherein z₁ is the first distance, r₂ is the second radius, M₀corresponds to a predetermined sustained high speed of the aircraft(such as cruise), and B2 is a value that ranges from 0.78 to 1.18.

Also, a second distance is defined between the forward-most end of thespinner and the axial location corresponding to the second radius, andthe ratio of the second distance to the second radius is less than4.115. Further the ratio of the second distance to the second radius maybe defined by EQ. 6:

$\frac{z_{2}}{r_{2}} = {C2\left( \frac{M_{0}}{{0.7}9} \right)^{3}}$

wherein z₂ is the second distance, r₂ is the second radius, M₀corresponds to a predetermined sustained high speed of the aircraft(such as cruise), and C2 is a value that ranges from 2.19 to 3.19.

An unducted propulsion system for an aircraft configured for highsubsonic cruise comprising: an axis of rotation; a forward bladeassembly comprised of a plurality of forward blades; an aft bladeassembly comprised of a plurality of aft blades; a forward housing; anaft housing; wherein each forward blade and each aft blade comprises ablade root proximal to the axis of rotation and a blade tip distal fromthe axis of rotation; wherein a flowpath curve corresponds to anintersection of the aft housing external surface with a plane containingthe axis of rotation and an aft-most point of an aft blade root; whereinfor the flowpath curve, an axial direction, z, is parallel to the axisof rotation and radius, r, is distance from the axis of rotation;wherein a bulge location with radius r_(b) on the flowpath curve isfound by proceeding aft from the aft-most point on the aft blade root towhere a first radius reaches a maximum; wherein a local minimum withradius r_(h) on the flowpath curve is found by proceeding forward fromthe bulge location to a nearest point where a second radius stopsdecreasing within an axial extent of the aft blade root; and whereinratio r_(b)/r_(h)>1.081.

The unducted propulsion system of any preceding clause wherein an axialdistance z_(b) is between the bulge location and the local minimum, andwherein ratio z_(b)/r_(h)<2.103.

The unducted propulsion system of any preceding clause wherein alocation with radius r_(m) is axially halfway between the bulge locationand the local minimum, and wherein

${{ratio}\frac{{r_{m}/r_{h}} - 1}{{r_{b}/r_{h}} - 1}} > {{0.5}{9.}}$

The unducted propulsion system of any preceding clause wherein theaircraft is configured for cruise flight Mach number 0.74<M₀<0.86, andwherein

${\frac{r_{b}}{r_{h}} = {{\left( {{A1} - 1} \right)\frac{M_{0} - 0.6}{{0.1}9}} + 1}},$where 1.11<A1<1.31.

The unducted propulsion system of any preceding clause wherein

${\frac{z_{b}}{r_{h}} = {B1\left( \frac{M_{0}}{{0.7}9} \right)^{3}}},$where 1.23<B1<1.63.

The unducted propulsion system of any preceding clause wherein

${\frac{{r_{m}/r_{h}} - 1}{{r_{b}/r_{h}} - 1} = {{0.5}9}},$where 0.59<C1<0.79.

The unducted propulsion system of any preceding clause wherein1.16<A1<1.31.

The unducted propulsion system of any preceding clause wherein1.23<B1<1.53.

The unducted propulsion system of any preceding clause wherein0.64<C1<0.79.

The unducted propulsion system of any preceding clause wherein1.16<A1<1.26.

The unducted propulsion system of any preceding clause wherein1.33<B1<1.53.

The unducted propulsion system of any preceding clause wherein0.64<C1<0.74.

The unducted propulsion system of any preceding clause wherein theaircraft is configured for dimensionless cruise thrust parameter,

$\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}},$where at cruise operation:

(i) F_(net) is fan net thrust,

(ii) ρ₀ is ambient air density,

(iii) V₀ is flight velocity,

(iv) A_(an) is fan stream tube annular area entering a fan; and

$\begin{matrix}{\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}} > {{0.0}6{0.}}} & (v)\end{matrix}$

The unducted propulsion system of any preceding clause wherein

$\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}} > {{0.0}8{0.}}$

The unducted propulsion system of any preceding clause wherein theforward blade assembly and the forward housing rotate about the axis ofrotation, and wherein the aft blade assembly and the aft housing arestationary.

The unducted propulsion system of any preceding clause wherein theflowpath curve further corresponds to respective aft-most points of twoor more aft blade roots.

The unducted propulsion system of any preceding clause wherein theflowpath curve further corresponds to respective aft-most points of atleast half of the aft blade roots.

The unducted propulsion system of any preceding clause wherein theforward blade assembly and the forward housing rotate about the axis ofrotation, wherein the aft blade assembly and a portion of the afthousing to which the plurality of aft blades are affixed rotate aboutthe axis of rotation, and wherein a third radius at a given axiallocation for which the aft housing is rotating is an effective radiusthat is a fourth radius of a circle having the same cross-sectional areaperpendicular to the axis of rotation at that axial location.

The unducted propulsion system of any preceding clause wherein theforward blade assembly and the forward housing are stationary, whereinthe aft blade assembly and a portion of the aft housing to which theplurality of aft blades are affixed rotate about the axis of rotation,and wherein a third radius at a given axial location for which the afthousing is rotating is an effective radius that is a fourth radius of acircle having the same cross-sectional area perpendicular to the axis ofrotation at that axial location.

The unducted propulsion system of any preceding clause wherein a numberof blades in the forward blade assembly is greater than 4, wherein anumber of blades in the aft blade assembly is greater than 4, andwherein a ratio of the number of blades in the forward blade assembly tothe number of blades in the aft blade assembly is between 2:5 and 2:1.

The unducted propulsion system of any preceding clause wherein a numberof blades in the forward blade assembly is between 8 and 18.

The unducted propulsion system of any preceding clause wherein adifference between the number of blades in the forward blade assembly tothe number of blades in the aft blade assembly is between 2 and −2.

The unducted propulsion system of any preceding clause wherein ratior_(b)/r_(h)>1.118.

The unducted propulsion system of any preceding clause wherein axialdistance z_(b) is between the bulge location and the local minimum, andwherein ratio z_(b)/r_(h)<1.974.

The unducted propulsion system of any preceding clause wherein alocation with radius r_(m) is axially halfway between the bulge locationand the local minimum, and wherein

${{ratio}\frac{{r_{m}/r_{h}} - 1}{{r_{b}/r_{h}} - 1}} > {{0.6}{4.}}$

The unducted propulsion system of any preceding clause wherein ratior_(b)/r_(h)<1.424.

An unducted propulsion system for an aircraft configured for highsubsonic cruise comprising: an axis of rotation; a forward bladeassembly comprised of a plurality of forward blades; an aft bladeassembly comprised of a plurality of aft blades; a forward housing; anaft housing; wherein for each forward blade and each aft blade comprisesa blade root proximal to the axis of rotation and a blade tip distalfrom the axis of rotation; wherein a flowpath curve corresponds to anintersection of the forward housing's external surface with a planecontaining the axis of rotation and a forward-most point of a forwardblade root; wherein for the flowpath curve, an axial direction, z, isparallel to the axis of rotation and radius, r, is a first distance fromthe axis of rotation; wherein a bulge location with radius r₁ on theflowpath curve is found by proceeding forward from the forward-mostpoint on the forward blade root to where a first radius reaches amaximum; wherein a local minimum with radius r₂ on the flowpath curve isfound by proceeding aft from the bulge location to a nearest point wherea second radius stops decreasing within an axial extent of the forwardblade root, and wherein ratio r₁/r₂>1.029.

The unducted propulsion system of any preceding clause wherein an axialdistance z₁ is between the bulge location and the local minimum, andwherein ratio z₁/r₂<1.522.

The unducted propulsion system of any preceding clause wherein an axialdistance z₂ is between a forward-most end of the forward housing and thelocal minimum, and wherein ratio z₂/r₂<4.115.

The unducted propulsion system of any preceding clause wherein theaircraft is configured for cruise flight Mach number 0.74<M₀<0.86, andwherein

${\frac{r_{1}}{r_{2}} = {{\left( {{A2} - 1} \right)\frac{M_{0} - 0.6}{0.19}} + 1}},$where 1.04<A2<1.14.

The unducted propulsion system of any preceding clause wherein

${\frac{z_{1}}{r_{2}} = {B2\left( \frac{M_{0}}{0.79} \right)^{3}}},$where 0.78<B2<1.18.

The unducted propulsion system of any preceding clause wherein

${\frac{z_{2}}{r_{2}} = {C2\left( \frac{M_{0}}{0.79} \right)^{3}}},$where 2.19<C2<3.19.

The unducted propulsion system of any preceding clause wherein1.06<A2<1.14.

The unducted propulsion system of any preceding clause wherein0.78<B2<1.08.

The unducted propulsion system of any preceding clause wherein2.19<C2<2.99.

The unducted propulsion system of any preceding clause wherein1.06<A2<1.12.

The unducted propulsion system of any preceding clause wherein0.88<B2<1.08.

The unducted propulsion system of any preceding clause wherein2.39<C2<2.99.

The unducted propulsion system of any preceding clause wherein theaircraft is configured for dimensionless cruise thrust parameter,

$\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}},$where at cruise operation:

(i) F_(net) is fan net thrust,

(ii) ρ₀ is ambient air density,

(iii) V₀ is flight velocity,

(iv) A_(an) is fan stream tube annular area entering a fan, and

$\begin{matrix}{\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}} > {0.06.}} & (v)\end{matrix}$

The unducted propulsion system of any preceding clause wherein

$\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}} > {0.08.}$

The unducted propulsion system of any preceding clause wherein theforward blade assembly and the forward housing are stationary, andwherein the aft blade assembly and a portion of the aft housing to whichthe plurality of aft blades are attached rotate about the axis ofrotation.

The unducted propulsion system of any preceding clause wherein theflowpath curve further corresponds to respective forward-most points oftwo or more forward blade roots.

The unducted propulsion system of any preceding clause wherein theflowpath curve further corresponds to respective forward-most points ofat least half of forward blade roots.

The unducted propulsion system of any preceding clause wherein theforward blade assembly and the forward housing rotate about the axis ofrotation, wherein the aft blade assembly and a portion of the afthousing to which the plurality of aft blades are affixed rotate aboutthe axis of rotation, and wherein a third radius at a given axiallocation for which the forward housing is rotating is an effectiveradius that is a fourth radius of a circle having the samecross-sectional area perpendicular to the axis of rotation at that axiallocation.

The unducted propulsion system of any preceding clause wherein theforward blade assembly and the forward housing rotate about the axis ofrotation, wherein the aft blade assembly and the aft housing arestationary, and wherein a third radius at a given axial location forwhich the forward housing is rotating is an effective radius that is afourth radius of a circle having the same cross-sectional areaperpendicular to the axis of rotation at that axial location.

The unducted propulsion system of any preceding clause wherein a numberof blades in the forward blade assembly is greater than 4, wherein anumber of blades in the aft blade assembly is greater than 4, andwherein a ratio of the number of blades in the forward blade assembly tothe number of blades in the aft blade assembly is between 2:5 and 2:1.

The unducted propulsion system of any preceding clause wherein thenumber of blades in the forward blade assembly is between 8 and 18.

The unducted propulsion system of any preceding clause wherein adifference between the number of blades in the forward blade assembly tothe number of blades in the aft blade assembly is between 2 and −2.

The unducted propulsion system of any preceding clause wherein ratior₁/r₂>1.044.

The unducted propulsion system of any preceding clause wherein an axialdistance z₁ is between the bulge location and the local minimum, andwherein ratio z₁/r₂<1.393.

The unducted propulsion system of any preceding clause wherein an axialdistance z₂ is between a forward-most end of the forward housing and thelocal minimum, and wherein ratio z₂/r₂<3.857.

The unducted propulsion system of any preceding clause wherein a span isa second distance between the blade root and the blade tip, wherein theplurality of forward blades in the forward blade assembly are orientedfor cruise operation, wherein the plurality of forward blades in theforward blade assembly have a maximum axial width near mid-span, andwherein 0 to 40 percent of the maximum axial width is located forward ofthe forward-most point of forward blade roots.

The unducted propulsion system of any preceding clause wherein aflowpath curve (z, r) coordinate system has origin at axial location ofthe local minimum with axial coordinate, z, increasing in a forwarddirection, wherein a curve forward of bulge (z>z₁) lies within lower andupper bounds defined by

${{\left( \frac{z - z_{1}}{z_{2} - z_{1}} \right)^{p} + \left( \frac{r}{r_{2}} \right)^{q}} = 1},$wherein a lower bound has exponents 1.5<p<2.0 and 2.0<q<3.0, and whereinan upper bound has exponents 2.0<p<3.0 and 3.0<q<3.5.

The unducted propulsion system of any preceding clause wherein the lowerbound has exponents 1.7<p<2.0 and 2.3<q<3.0, and wherein the upper boundhas exponents 2.0<p<2.5 and 3.0<q<3.3.

An unducted propulsion system for an aircraft configured for highsubsonic cruise comprising: an axis of rotation; a forward bladeassembly comprised of a plurality of forward blades; a forward housing;wherein for each blade comprises a blade root proximal to the axis ofrotation and a blade tip distal from the axis of rotation; wherein aflowpath curve corresponds to an intersection of the forward housing'sexternal surface with a plane containing the axis of rotation and aforward-most point of a forward blade root; wherein for the flowpathcurve, an axial direction, z, is parallel to the axis of rotation andradius, r, is a distance from the axis of rotation; wherein a bulgelocation with radius r₁ on the flowpath curve is found by proceedingforward from the forward-most point on the forward blade root to where afirst radius reaches a maximum; wherein a local minimum with radius r₂on the flowpath curve is found by proceeding aft from the bulge locationto a nearest point where a second radius stops decreasing, and whereinratio r₁/r₂>1.066.

The unducted propulsion system of any preceding clause wherein an axialdistance z₁ is between the bulge location and the local minimum, andwherein ratio z₁/r₂<1.522.

The unducted propulsion system of any preceding clause wherein an axialdistance z₂ is between a forward-most end of the forward housing and thelocal minimum, and wherein ratio z₂/r₂<4.115.

What is claimed is:
 1. An unducted propulsion system for an aircraftconfigured for high subsonic cruise comprising: an axis of rotation; aforward blade assembly comprised of a plurality of forward blades; anaft blade assembly comprised of a plurality of aft blades; a forwardhousing; an aft housing; wherein each forward blade comprises a forwardblade root proximal to the axis of rotation and a forward blade tipdistal from the axis of rotation; wherein each aft blade comprises anaft blade root proximal to the axis of rotation and an aft blade tipdistal from the axis of rotation; wherein a flowpath curve correspondsto an intersection of an external surface of the aft housing with aplane containing the axis of rotation and an aft-most point of the aftblade root; wherein for the flowpath curve, an axial direction, z, isparallel to the axis of rotation and radius, r, is distance from theaxis of rotation; wherein a bulge location with radius r_(b) on theflowpath curve is found by proceeding aft from the aft-most point on theaft blade root to where a first radius reaches a maximum; wherein alocal minimum with radius r_(h) on the flowpath curve is found byproceeding forward from the bulge location to a nearest point where asecond radius stops decreasing within an axial extent of the aft bladeroot; wherein ratio r_(b)/r_(h)>1.081; wherein the ratior_(b)/r_(h)<1.424; and wherein the aircraft is configured for cruiseflight Mach number 0.74<M₀<0.86, and wherein${\frac{r_{b}}{r_{h}} = {{\left( {{A1} - 1} \right)\frac{M_{0} - 0.6}{0.19}} + 1}},$where 1.11<A1<1.31.
 2. The unducted propulsion system of claim 1,wherein an axial distance z_(b) is between the bulge location and thelocal minimum, and wherein ratio z_(b)/r_(h)<2.103.
 3. The unductedpropulsion system of claim 2, wherein a location with radius r_(m) isaxially halfway between the bulge location and the local minimum, andwherein ${{ratio}\frac{{r_{m}/r_{h}} - 1}{{r_{b}/r_{h}} - 1}} > {0.59.}$4. The unducted propulsion system of claim 1, wherein${\frac{z_{b}}{r_{h}} = {B1\left( \frac{M_{0}}{0.79} \right)^{3}}},$where 1.23<B1<1.63.
 5. The unducted propulsion system of claim 4,wherein ${\frac{{r_{m}/r_{h}} - 1}{{r_{b}/r_{h}} - 1} = {C1}},$ where0.59<C1<0.79.
 6. The unducted propulsion system of claim 5, wherein1.16<A1<1.31.
 7. The unducted propulsion system of claim 6, wherein1.23<B1<1.53.
 8. The unducted propulsion system of claim 7, wherein0.64<C1<0.79.
 9. The unducted propulsion system of claim 8, wherein1.16<A1<1.26.
 10. The unducted propulsion system of claim 9, wherein1.33<B1<1.53.
 11. The unducted propulsion system of claim 10, wherein0.64<C1<0.74.
 12. The unducted propulsion system of claim 5, wherein theaircraft is configured for dimensionless cruise thrust parameter,$\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}},$ where at cruise operation i.F_(net) is fan net thrust, ii. ρ₀ is ambient air density, iii. V₀ isflight velocity, iv. A_(an) is fan stream tube annular area entering afan; and v. $\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}} > {0.06.}$
 13. Theunducted propulsion system of claim 12, wherein$\frac{F_{net}}{\rho_{0}V_{0}^{2}A_{an}} > {0.08.}$
 14. The unductedpropulsion system of claim 5, wherein the forward blade assembly and theforward housing rotate about the axis of rotation, and wherein the aftblade assembly and the aft housing are stationary.
 15. The unductedpropulsion system of claim 14, wherein the flowpath curve furthercorresponds to respective aft-most points of two or more aft bladeroots.
 16. The unducted propulsion system of claim 14, wherein theflowpath curve further corresponds to respective aft-most points of atleast half of the aft blade roots.
 17. The unducted propulsion system ofclaim 5, wherein the forward blade assembly and the forward housingrotate about the axis of rotation, wherein the aft blade assembly and aportion of the aft housing to which the plurality of aft blades areaffixed rotate about the axis of rotation, and wherein a third radius ata given axial location for which the aft housing is rotating is aneffective radius that is a fourth radius of a circle having the samecross-sectional area perpendicular to the axis of rotation at the givenaxial location.
 18. The unducted propulsion system of claim 5, whereinthe forward blade assembly and the forward housing are stationary,wherein the aft blade assembly and a portion of the aft housing to whichthe plurality of aft blades are affixed rotate about the axis ofrotation, and wherein a third radius at a given axial location for whichthe aft housing is rotating is an effective radius that is a fourthradius of a circle having the same cross-sectional area perpendicular tothe axis of rotation at the given axial location.
 19. The unductedpropulsion system of claim 5, wherein a number of blades in the forwardblade assembly is greater than 4, wherein a number of blades in the aftblade assembly is greater than 4, and wherein a ratio of the number ofblades in the forward blade assembly to the number of blades in the aftblade assembly is between 2:5 and 2:1.
 20. The unducted propulsionsystem of claim 19, wherein a number of blades in the forward bladeassembly is between 8 and
 18. 21. The unducted propulsion system ofclaim 20, wherein a difference between the number of blades in theforward blade assembly to the number of blades in the aft blade assemblyis between 2 and −2.
 22. The unducted propulsion system of claim 1,wherein the ratio r_(b)/r_(h)>1.118.
 23. The unducted propulsion systemof claim 22, wherein axial distance z_(b) is between the bulge locationand the local minimum, and wherein ratio z_(b)/r_(h)<1.974.
 24. Theunducted propulsion system of claim 23, wherein a location with radiusr_(m) is axially halfway between the bulge location and the localminimum, and wherein${{ratio}\frac{{r_{m}/r_{h}} - 1}{{r_{b}/r_{h}} - 1}} > {0.64.}$